With the large bulk of baby boomers getting closer to their golden years, a great deal of public interest is shifting toward how individuals should finance their retirement. Although a substantial portion of retirement needs is usually provided by company and government pensions, there is often a large gap that can only be funded with discretionary savings. Indeed, based on the Survey of Consumer Finances, the number one reason for households to save is to finance retirement.
One of the most important decisions investors have to make is how to allocate their retirement savings to finance their retirement consumption. Investors face two major risk factors when making asset allocation decisions in retirement: financial market risk and longevity risk. Financial market risk is the performance uncertainty of investment vehicles. It is typically measured and quantified by the volatility of the investment returns. Longevity risk, on the other hand, is the uncertainty of how long the consumer will need retirement income (i.e., the risk that the consumer could potentially outlive his or her investments). Financial market risk is related to financial market behavior and economic the environment; longevity risk is normally related to the health condition of the investor.
Traditionally, asset allocation is determined by constructing efficient portfolios for various risk levels based on modern portfolio theory (MPT) developed by Harry Markowitz (1952) “Portfolio Selection,” Journal of Finance, September 1952, pp.77–91; and later by Sharpe (1965) “Capital Asset Prices: A Theory of Market Equilibrium,” Journal of Finance, September 1964. According to the investor's risk tolerance, one of the efficient portfolios is chosen. MPT is widely accepted in the academic and the finance industry as the primary tool for developing asset allocations. However, two important factors not considered in MPT make its effectiveness questionable when dealing with asset allocations for individual investors in retirement. First, longevity risk is not considered. Second, portfolios in retirement often face periodical withdrawals to finance retirement consumption. Being a one-period model, MPT does not take periodic withdrawals into consideration.
In “Optimum Consumption and Portfolio Rules in a Continuous-Time Model”, Journal of Economic Theory, (1971) Vol. 3, pp. 373–413, and in “Optimal Consumption, Portfolio and Life Insurance Rules for an Uncertain Lived Individual in a Continuous Time Model”, The Journal of Financial Economics, (1975) Vol. 2, pp. 187–203, a multi-period extension of MPT which incorporates insurance is disclosed. However, even this extension of MPT fails to account for the specifics of a desired retirement standard of living and the irreversibility of immediate annuities.
Annuitization involves paying a non-refundable lump sum to an insurance company. In exchange, the insurance company guarantees a constant life-long payment stream that can not be outlived, but is terminated upon death. The generic life payout annuity provides no bequest for heirs, although spousal protection, in the form of joint-and-last-survivor, can be purchased at additional cost. Annuities can be viewed as longevity insurance, a type of mortality-contingent claim.
To create the contract, the annuity (insurance) company receives money from each person who buys a payout annuity. The funds received from each person are pooled into a collective pool from which claim payments are made. An insured person's annuity claim is contingent on continued life. The annuity payments are calculated so that each recipient gets the largest amount that the company can pay without exceeding the fund created from the accumulations of the entire group. The funds from people who die earlier than average support those who live longer than average. Many people purchase life annuities in order to be assured of an income they can't outlive. Others hesitate to select a lifetime payout annuity because they must give up control of the funds and hence the amount they can pass to their heirs. Just like other types of insurance, the amount and kind of immediate annuities an investor should own depends on many factors, including: (i) how risk averse he or she is; (ii) how much wealth has been accumulated; (iii) how much is the retirement need; (iv) how long he or she needs the savings to last; and (v) how strongly he or she feels about bequest. The fees and costs of annuity investments relative to other investment vehicles (such as mutual funds) as well as the tax structure and the investor's tax bracket also have impacts on the optimal allocations.
Despite the importance of this decision to each and every retiree, there have been relatively few scholarly papers written or models developed on the normative topic of the optimal allocation to immediate annuities. Most of the literature tends to focus on a positive economic analysis of the immediate annuity market.
For example, a substantial body of literature has documented the extremely low levels of voluntary annuitization exhibited amongst elderly retirees. Strictly speaking, this phenomenon is inconsistent with results of the life-cycle model of savings and consumption disclosed by M. Yaari in “Uncertain Lifetime, Life Insurance and the Theory of the Consumer”, Review of Economic Studies, Vol. 32, pp. 137–150, (1965). Yaari examines a standard Ando & Modigliani life-cycle model of savings and consumption with no bequest motives, and demonstrates that all consumers hold “actuarial notes” (immediate annuities) as opposed to liquid assets. Yaari's study implies that when given the chance, retirees should convert their liquid assets to life annuities, which provide longevity insurance and protection against outliving one's money. The rationale behind Yaari's result is that returns from actuarial notes (life annuities) dominate all other assets because the “living” inherit the assets and returns of the “dead”. Moreover, at older ages, the higher probability of dying increases the relative return, conditional on survival, from actuarial notes.
Nevertheless, despite the highly appealing arguments in favor of annuitization, there is little evidence that retirees are voluntarily embracing this arrangement. Notably, very few people consciously choose to annuitize their marketable wealth. In the comprehensive Health and Retirement Survey (HRS), conducted in the U.S, only 1.57% of the HRS respondents reported annuity income. Likewise, only 8.0% of respondents with a defined contribution pension plan selected an annuity payout. The U.S. based Society of Actuaries, conducted a study that shows that less than 1% of variable annuity (VA) contracts were annuitized during the 1992–1994 period.
Brugiavini discloses a conventional model in “Uncertainty Resolution and the Timing of Annuity Purchases”, Journal of Public Economics, Vol. 50, pp. 31–62, (1993). Brugiavini examined the optimal time to annuitize, and concludes that it should be early in the life cycle. However, Brugiavini's model assumes that assets earns the same risk-free rate of return, and does not examine the best asset mix for annuities. In a similar vein, Blake, Cairns and Dowd (2000) “PensionMetrics: Stochastic Pension Plan Design during the Distribution Phase”, Pensions Institute Working Paper conducted extensive computer simulations to determine the annuity and pension draw down policy that provides the highest level of (exponential) utility. However, Blake et al. did not examine the implications of annuitizing at different ages, as it pertains to the best time to annuitize, or the best asset mix within the annuity.
In related research, S. Kapur and J. M. Orszag introduced immediate annuities into a Merton (1971) framework in “A Portfolio Approach to Investment and Annuitization During Retirement”, Birbeck College (University of London) Mimeo, May 1999. This study assumed that the risk-free rate is augmented by a mortality bonus that is proportional to the instantaneous hazard rate. However, this study ignored variable immediate annuities and the irreversibility of the annuity contract. As such, their results are difficult to apply in a portfolio context.
One-Period Model
A life payout annuity is a financial investment whose returns are enhanced by pooling mortality risk with others. Here is a simple example of a one-period life annuity that illustrates the concept of risk pooling. According to U.S. life tables compiled by the Center for Disease Control and Prevention (FIG. 1), there is a 20% chance that any given 95-year-old (white) female will die during the next year. In other words, for any given large group of 95-year-old females, 20% will not survive for another year. Of course, it is not possible to determine which 20% will not survive.
Now, imagine that five such 95 year-old females entered into the following legally binding agreement. Each of the five females has agreed to contribute $100 to a communal fund that will invest in Treasury bills yielding 5%. Then, according to the contract, at the end of the year, only the surviving females will be entitled to split the proceeds of the fund.
Clearly, the total contribution of $500 will grow to $525 by the end of the year. And, if all five females are still alive—they are now 96 years old—they will each receive $105. This is precisely the $100 investment, plus interest. If one of them dies during the next year, the remaining four will be entitled to split the $525, giving each a total of $131.25. Recall that the agreement stipulated that those who die cede control of their assets. The four survivors will therefore gain a return of 31.25% on their money. In fact, if two happen to die, the remaining three will get $175, which is an impressive 75% return on their money. In other words, the survivors' gains are comprised of their original principal, their interest, and other people's principal and interest. By pooling mortality risk and ceding bequests, everyone gains.
Technically, this agreement is called a tontine, also known as a pure endowment contract which will be referred to hereinafter as a one-period life annuity contract.
Of course, with only five females in the mortality pool, six different things can happen. In the two extreme cases, they all might die, or they all might survive. But with 10,000 such females entering into a one-period annuity agreement, it is pretty much assured that the $1,050,000 will be split among 8,000 survivors. In other words, the expected return from the contract for the survivors is (1,050,000/8,000)=$131.25, or a 31.25% gain. The numerator is the total return for the pool, and the denominator represents the survivors.
Algebraically, if R denotes the risk-free interest rate (U.S. Treasury Bills) per period, and p is the probability of survival per period, then the return from the one-period annuity is (1+R)/p−1>R, where p denotes the probability of survival. This is the return for the survivors. And again, the reason the return is greater than R is because the dead subsidize the living. Furthermore, the smaller the p, the greater is the 1/p and the greater is the return from the one-period life annuity.
Now, this arrangement is not as outrageous or artificial as it may first sound. In fact, it is the principle underlying all immediate annuities, and all pension plans for that matter. In practice, the agreement is made over a series of periods, as opposed to just one. But the mechanics are the same, and the survivors derive a higher return compared to placing their funds in a conventional asset (non-mortality contingent) asset.
Although the example we have provided assumes that R (the risk-free interest rate) is fixed, the same would apply for a Variable and unknown return denoted by X. Moreover, the exact same principle would apply with a variable investment return as well. In fact, the returns might be even higher. Namely, the 10,000 females can invest their $100 in a stock mutual fund that earns 5%, or 10% or even 20%. They do not know in advance, what the fund/pool will earn. At the end of the year, the annuitants will learn (or realize) their investment returns, and then split the gains among the surviving pool. Moreover, in the event the investment earns a negative return, i.e., loses money, the participants will share in the losses as well, but the effect will be mitigated by the mortality credits.
If five females invest $100 in a fund that earns a random return, the expected return for the survivors is (1+X)/p−1>X. This concept is the foundation of a variable immediate annuity, which is the counterpart to a fixed immediate annuity.
Both fixed and variable immediate annuities provide longevity insurance via the mortality credits but with fixed annuities the pool has been invested in fixed income securities, and with variable annuities the pool has been invested in variable return (read stocks, real estate, etc.) securities. The choices and decisions made between fixed and variable immediate annuities are identical to the choices between fixed and variable accumulation products. The optimal allocations should depend on the participant's risk aversion, comfort with the fluctuating stock market, time horizon, and budgeting requirements.
As such, the fixed immediate annuity is one particular asset class within the portfolio of longevity insurance products. Thus, providing a fixed immediate annuity without access to a variable immediate annuity is akin to offering a money market and bond fund in a (pension) savings plan without providing equity-based products to span the risk and return spectrum.
In practice, only insurance-chartered companies are authorized to provide these mortality-contingent products. In fact, most insurance companies go one step further and actually guarantee that you will receive the mortality credit enhancements, even if the mortality experience of the participants is better than expected. In other words, in the above-mentioned example, with an expected 20% mortality rate, they would guarantee that all survivors receive 31.25% on their money, regardless of whether or not 20% of the group died during the year.
Insurance companies are able to provide this guarantee by making very careful and conservative assumptions about the rate of return earned on assets. Furthermore, the greater the number of insurance annuities an insurance company sells, or has on its books, the lower the risk is in providing this guarantee. These are the ultimate economics of scale. In other words, the risks might be significant if they only sold five such policies, but with 500,000 policies, the probability of an adverse outcome is close to zero.